One in every of my daughters gave me a Klein bottle for Christmas.
Think about beginning with a cylinder and becoming a member of the 2 ends collectively. This makes a torus (doughnut). However if you happen to twist the ends earlier than becoming a member of them, very similar to you twist the ends of an oblong strip to make a Möbius strip, you get a Klein bottle. This isn’t attainable to do in 3D with out making the cylinder go via itself, so that you’re purported to think about that the half the place the bottle intersects itself isn’t there.
However is a Klein bottle actual? My Christmas current is an actual bodily object, so it’s actual in that sense. Is a Klein bottle actual as a mathematical object? Can or not it’s outlined with none enchantment to imagining issues that aren’t true? Sure it may well.
Formal definition
Begin with a unit sq., the set of factors (x, y) with 0 ≤ x, y ≤ 1. When you establish the highest and backside of the sq., the factors with y coordinate equal to 0 or 1, you get a cylinder. You possibly can think about curling the sq. in 3D and taping the highest and backside collectively.
Equally, if you happen to begin with the unit sq. and establish the vertical sides along with a twist, you get a Möbius strip. You received’t have the ability to bodily do that with a sq., however you possibly can with a rectangle. Or you possibly can think about the sq. to be made out of rubber, and also you stretch it earlier than you twist it and be part of the perimeters collectively.
When you begin with the unit sq. and do each issues described above—be part of the highest and backside as-is and be part of the edges with a twist—you get a Klein bottle. You possibly can’t fairly bodily do each on the similar time in 3D; you’d have to chop just a little gap within the sq. to let a part of the sq. go via, as within the glass bottle on the high of the put up.
Though you may’t assemble a bodily Klein bottle and not using a little bit of dishonest, there’s nothing improper with the mathematical definition. There are some particulars which were neglected, however there’s nothing unlawful concerning the building.
Extra formality
To fill within the lacking particulars, we now have to say simply what we imply by figuring out factors. Once we establish the highest edge and backside fringe of the sq. to make a cylinder, we imply that we think about that for each x, (x, 0) and (x, 1) are the identical level. Equally, once we establish the edges with a twist, we think about that for all y, (0, y) and (1, 1 − y) are the identical level.
However that is unsatisfying. What does all this imagining imply? How is that this any higher than imagining that the opening within the glass bottle isn’t there? We will outline what it means to “establish” or “glue” edges collectively in a manner that’s completely rigorous.
We will say that as a set of factors, the Klein bottle is
Ok = [0, 1) × [0, 1),
removing the top and right edge. But what makes this set of points a Klein bottle is the topology we put on it, the way we define which points are close together.
We define an ε neighborhood of a point (x, 0) to be the union of two half disks, the intersection with K of an open disk of radius ε centered at (x, 0) and the intersection with K of an open disk centered at (x, 1). This is a way to make rigorous the idea of gluing (x, 0) and (x, 1) together.
Along the same lines, we define an ε neighborhood of a point (0, y) to be the intersection with K of an open disk of radius ε centered at (0, y) and an open disk of radius ε centered at (1, 1 − y).
The discussion with coordinates is more complicated than the talk about imagining this and that, but it’s more rigorous. You can’t have simplicity and rigor at the same time, so you alternate back and forth. You think in terms of the simple visualization, but when you’re concerned that you may be saying something untrue, you go down to the detail of coordinates and prove things carefully.
Topology can seem all hand-wavy because that’s how topologist communicate. They speak in terms of twisting this and glueing that. But they have in the back of their mind that all these manipulations can be justified. The formalism may be left implicit, even in a scholarly publication, when it’s assumed that the reader could fill in the details. But when things are more subtle, the formalism is written out.
Escaping 3D
In the construction above, we define the Klein bottle as a set of points in the 2D plane with a new topology. That works, but there’s another approach. I said above that you can’t join the edges to make a Klein bottle in three dimensions. I added this disclaimer because you can join the edges without cheating if you work in higher dimensions.
If you’d like a parameterization of the Klein bottle, say because you want to calculate something, you can do that, but you’ll need to work in four dimensions. There’s more room to move around in higher dimensions, letting you do things you can’t do in three dimensions.
